This is an accompaniment to Lecture 8. More on Linear Regression.

library(foreign) #To work with SPSS data
library(lmSupport)#Extra functions for linear model (may require install of nloptr also)
library(lm.beta)
library(stargazer)#pretty print regression output
## 
## Please cite as:
##  Hlavac, Marek (2018). stargazer: Well-Formatted Regression and Summary Statistics Tables.
##  R package version 5.2.2. https://CRAN.R-project.org/package=stargazer
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(ppcor)
## Loading required package: MASS
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
library(car)
## Loading required package: carData
## 
## Attaching package: 'car'
## The following object is masked from 'package:dplyr':
## 
##     recode
#We are using our exam result dataset - regression.sav
#Read in the file
regression<- read.spss("regression.sav", use.value.labels=TRUE, max.value.labels=Inf, to.data.frame=TRUE)

First Model NormExam predicted by StandLRT

model1<-lm(regression$normexam~regression$standlrt)
anova(model1)
## Analysis of Variance Table
## 
## Response: regression$normexam
##                       Df Sum Sq Mean Sq F value    Pr(>F)    
## regression$standlrt    1 1417.5 1417.50    2185 < 2.2e-16 ***
## Residuals           4057 2631.9    0.65                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(model1)
## 
## Call:
## lm(formula = regression$normexam ~ regression$standlrt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.65615 -0.51848  0.01264  0.54399  2.97399 
## 
## Coefficients:
##                      Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         -0.001191   0.012642  -0.094    0.925    
## regression$standlrt  0.595057   0.012730  46.744   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8054 on 4057 degrees of freedom
## Multiple R-squared:   0.35,  Adjusted R-squared:  0.3499 
## F-statistic:  2185 on 1 and 4057 DF,  p-value: < 2.2e-16
lm.beta(model1)
## 
## Call:
## lm(formula = regression$normexam ~ regression$standlrt)
## 
## Standardized Coefficients::
##         (Intercept) regression$standlrt 
##           0.0000000           0.5916496
stargazer(model1, type="text") #Tidy output of all the required stats
## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                              normexam          
## -----------------------------------------------
## standlrt                     0.595***          
##                               (0.013)          
##                                                
## Constant                      -0.001           
##                               (0.013)          
##                                                
## -----------------------------------------------
## Observations                   4,059           
## R2                             0.350           
## Adjusted R2                    0.350           
## Residual Std. Error      0.805 (df = 4057)     
## F Statistic         2,185.011*** (df = 1; 4057)
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

Second model including dummy variable for gender

model2<-lm(regression$normexam~regression$standlrt+regression$girl)
anova(model2)
## Analysis of Variance Table
## 
## Response: regression$normexam
##                       Df  Sum Sq Mean Sq  F value    Pr(>F)    
## regression$standlrt    1 1417.50 1417.50 2208.010 < 2.2e-16 ***
## regression$girl        1   28.06   28.06   43.704 4.317e-11 ***
## Residuals           4056 2603.88    0.64                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(model2)
## 
## Call:
## lm(formula = regression$normexam ~ regression$standlrt + regression$girl)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.56172 -0.51893  0.01808  0.53604  2.90399 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         -0.10318    0.01990  -5.184 2.28e-07 ***
## regression$standlrt  0.59060    0.01268  46.571  < 2e-16 ***
## regression$girlgirl  0.16996    0.02571   6.611 4.32e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8012 on 4056 degrees of freedom
## Multiple R-squared:  0.357,  Adjusted R-squared:  0.3567 
## F-statistic:  1126 on 2 and 4056 DF,  p-value: < 2.2e-16
stargazer(model2, type="text") #Tidy output of all the required stats
## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                              normexam          
## -----------------------------------------------
## standlrt                     0.591***          
##                               (0.013)          
##                                                
## girlgirl                     0.170***          
##                               (0.026)          
##                                                
## Constant                     -0.103***         
##                               (0.020)          
##                                                
## -----------------------------------------------
## Observations                   4,059           
## R2                             0.357           
## Adjusted R2                    0.357           
## Residual Std. Error      0.801 (df = 4056)     
## F Statistic         1,125.857*** (df = 2; 4056)
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01
stargazer(model1, model2, type="text") #Quick model comparison
## 
## ===========================================================================
##                                       Dependent variable:                  
##                     -------------------------------------------------------
##                                            normexam                        
##                                 (1)                         (2)            
## ---------------------------------------------------------------------------
## standlrt                     0.595***                    0.591***          
##                               (0.013)                     (0.013)          
##                                                                            
## girlgirl                                                 0.170***          
##                                                           (0.026)          
##                                                                            
## Constant                      -0.001                     -0.103***         
##                               (0.013)                     (0.020)          
##                                                                            
## ---------------------------------------------------------------------------
## Observations                   4,059                       4,059           
## R2                             0.350                       0.357           
## Adjusted R2                    0.350                       0.357           
## Residual Std. Error      0.805 (df = 4057)           0.801 (df = 4056)     
## F Statistic         2,185.011*** (df = 1; 4057) 1,125.857*** (df = 2; 4056)
## ===========================================================================
## Note:                                           *p<0.1; **p<0.05; ***p<0.01

Third model including interaction term (gender by standlrt)

model3<-lm(regression$normexam~regression$standlrt+regression$girl+regression$interaction)
stargazer(model3, type="text") #Tidy output of all the required stats
## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                              normexam          
## -----------------------------------------------
## standlrt                     0.593***          
##                               (0.019)          
##                                                
## girlgirl                     0.170***          
##                               (0.026)          
##                                                
## interaction                   -0.004           
##                               (0.025)          
##                                                
## Constant                     -0.103***         
##                               (0.020)          
##                                                
## -----------------------------------------------
## Observations                   4,059           
## R2                             0.357           
## Adjusted R2                    0.357           
## Residual Std. Error      0.801 (df = 4055)     
## F Statistic          750.399*** (df = 3; 4055) 
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01
stargazer(model2, model3, type="text") #Quick model comparison
## 
## =========================================================================
##                                      Dependent variable:                 
##                     -----------------------------------------------------
##                                           normexam                       
##                                 (1)                        (2)           
## -------------------------------------------------------------------------
## standlrt                     0.591***                   0.593***         
##                               (0.013)                    (0.019)         
##                                                                          
## girlgirl                     0.170***                   0.170***         
##                               (0.026)                    (0.026)         
##                                                                          
## interaction                                              -0.004          
##                                                          (0.025)         
##                                                                          
## Constant                     -0.103***                  -0.103***        
##                               (0.020)                    (0.020)         
##                                                                          
## -------------------------------------------------------------------------
## Observations                   4,059                      4,059          
## R2                             0.357                      0.357          
## Adjusted R2                    0.357                      0.357          
## Residual Std. Error      0.801 (df = 4056)          0.801 (df = 4055)    
## F Statistic         1,125.857*** (df = 2; 4056) 750.399*** (df = 3; 4055)
## =========================================================================
## Note:                                         *p<0.1; **p<0.05; ***p<0.01

Fourth model including dummy variables for school gender

model4<-lm(regression$normexam~regression$standlrt+regression$girl+regression$boys_sch+regression$girls_sch)
stargazer(model4, type="text") #Tidy output of all the required stats
## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                              normexam          
## -----------------------------------------------
## standlrt                     0.591***          
##                               (0.013)          
##                                                
## girlgirl                     0.133***          
##                               (0.034)          
##                                                
## boys_sch                     0.183***          
##                               (0.043)          
##                                                
## girls_sch                    0.168***          
##                               (0.033)          
##                                                
## Constant                     -0.161***         
##                               (0.024)          
##                                                
## -----------------------------------------------
## Observations                   4,059           
## R2                             0.364           
## Adjusted R2                    0.363           
## Residual Std. Error      0.797 (df = 4054)     
## F Statistic          580.148*** (df = 4; 4054) 
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01

FULL Example from SURVEY.dat

library(userfriendlyscience)
library(ppcor)
library(olsrr)
## 
## Attaching package: 'olsrr'
## The following object is masked from 'package:MASS':
## 
##     cement
## The following object is masked from 'package:datasets':
## 
##     rivers
#We are using a .dat file (survey.dat) created from the SPSS file survey.sav  taken from SPSS Survival Manual 6th Edition Julie Pallant
#http://spss.allenandunwin.com.s3-website-ap-southeast-2.amazonaws.com/data-files.html#.Wb0vvnWP-po
#Results on a survey on well being
#We need to load the file so that we can use it in R.
sdata <- read.table("survey.dat")
#Setting the column names to be that used in the dataset
colnames(sdata) <- tolower(colnames(sdata))






#Look first at partial correlation
#First fiter out variables of interest and filter out na
myvars <- c("tpcoiss", "tpstress", "toptim", "tmarlow")
ydata <-na.omit(sdata[myvars])


#perception of control and stress controlling for social desirability
ppcor::spcor.test(ydata$tpcoiss, ydata$tpstress, ydata$tmarlow)
##     estimate      p.value statistic   n gp  Method
## 1 -0.5269534 1.862222e-31 -12.69154 422  1 pearson
#perception of control and optimism controlling for social desirability
ppcor::spcor.test(ydata$tpcoiss, ydata$toptim, ydata$tmarlow)
##    estimate      p.value statistic   n gp  Method
## 1 0.4830465 5.348979e-26  11.29257 422  1 pearson
#stress and optimism controlling for social desirability
ppcor::spcor.test(ydata$tpstress, ydata$toptim, ydata$tmarlow)
##     estimate     p.value statistic   n gp  Method
## 1 -0.4412444 1.73877e-21 -10.06483 422  1 pearson
#Get zero order correlations as well to fully explore the effect 
cor.test(ydata$tpcoiss, ydata$tpstress)
## 
##  Pearson's product-moment correlation
## 
## data:  ydata$tpcoiss and ydata$tpstress
## t = -14.639, df = 420, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.6411275 -0.5143192
## sample estimates:
##        cor 
## -0.5812413
cor.test(ydata$tpcoiss, ydata$toptim)
## 
##  Pearson's product-moment correlation
## 
## data:  ydata$tpcoiss and ydata$toptim
## t = 12.351, df = 420, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4425181 0.5829097
## sample estimates:
##       cor 
## 0.5161727
cor.test(ydata$tpstress, ydata$toptim)
## 
##  Pearson's product-moment correlation
## 
## data:  ydata$tpstress and ydata$toptim
## t = -10.816, df = 420, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.5382333 -0.3886119
## sample estimates:
##        cor 
## -0.4667559
#Look at differences in scores for gender
#Conduct Levene's test for homogeneity of variance in library car
leveneTest(tpstress ~ sex, data=sdata)
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group   1  1.9363 0.1648
##       431
#Conduct the t-test
#You can use the var.equal = TRUE option to specify equal variances and a pooled variance estimate
t.test(tpstress~sex,var.equal=FALSE,data=sdata)#Variances are not equal
## 
##  Welch Two Sample t-test
## 
## data:  tpstress by sex
## t = 2.948, df = 415.89, p-value = 0.003379
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.5443502 2.7229363
## sample estimates:
## mean in group FEMALES   mean in group MALES 
##              27.42169              25.78804
#Conduct Levene's test for homogeneity of variance in library car
leveneTest(tpcoiss ~ sex, data=sdata)
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group   1  2.6386  0.105
##       428
#Conduct the t-test
#You can use the var.equal = TRUE option to specify equal variances and a pooled variance estimate
t.test(tpcoiss~sex,var.equal=FALSE,data=sdata)
## 
##  Welch Two Sample t-test
## 
## data:  tpcoiss by sex
## t = -2.0357, df = 410.38, p-value = 0.04243
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -4.59066957 -0.08019182
## sample estimates:
## mean in group FEMALES   mean in group MALES 
##              59.63710              61.97253
#Conduct Levene's test for homogeneity of variance in library car
leveneTest(toptim ~ sex, data=sdata)
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value  Pr(>F)  
## group   1  4.2058 0.04089 *
##       433                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#Conduct the t-test
#You can use the var.equal = TRUE option to specify equal variances and a pooled variance estimate
t.test(toptim~sex,var.equal=TRUE,data=sdata)
## 
##  Two Sample t-test
## 
## data:  toptim by sex
## t = 0.42849, df = 433, p-value = 0.6685
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.6612571  1.0299563
## sample estimates:
## mean in group FEMALES   mean in group MALES 
##              22.19522              22.01087
#Conduct Levene's test for homogeneity of variance in library car
leveneTest(tmarlow ~ sex, data=sdata)
## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group   1  0.0132 0.9087
##       431
#Conduct the t-test
#You can use the var.equal = TRUE option to specify equal variances and a pooled variance estimate
t.test(tmarlow~sex,var.equal=FALSE,data=sdata)
## 
##  Welch Two Sample t-test
## 
## data:  tmarlow by sex
## t = 2.2975, df = 398.33, p-value = 0.02211
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.06526408 0.83930423
## sample estimates:
## mean in group FEMALES   mean in group MALES 
##              5.496000              5.043716
#install the library userfriendlyscience and load it using the library command
#it has a really nice one-way anova function that provides
#nice summary output


#run a one-way anova test using the correct post-hoc test Tukey in our case
#Use Games-Howell for unequal variances
one.way <- oneway(sdata$agegp3, y = sdata$toptim, posthoc = 'Tukey') 
#printout a summary of the anova 
one.way 
## ### Oneway Anova for y=toptim and x=agegp3 (groups: 18 - 29, 30 - 44, 45+)
## 
## Omega squared: 95% CI = [0; .05], point estimate = .02
## Eta Squared: 95% CI = [0; .05], point estimate = .02
## 
##                                      SS  Df    MS    F   p
## Between groups (error + effect)  179.07   2 89.53 4.64 .01
## Within groups (error only)      8333.95 432 19.29         
## 
## 
## ### Post hoc test: Tukey
## 
##                 diff lwr   upr  p adj
## 30 - 44-18 - 29 0.74 -0.45 1.94 .308 
## 45+-18 - 29     1.6  0.36  2.83 .007 
## 45+-30 - 44     0.85 -0.37 2.07 .230
one.way <- oneway(sdata$agegp3, y = sdata$tpstress, posthoc = 'Tukey') 
#printout a summary of the anova 
one.way
## ### Oneway Anova for y=tpstress and x=agegp3 (groups: 18 - 29, 30 - 44, 45+)
## 
## Omega squared: 95% CI = [NA; .03], point estimate = .01
## Eta Squared: 95% CI = [0; .03], point estimate = .01
## 
##                                      SS  Df    MS    F    p
## Between groups (error + effect)  159.95   2 79.97 2.35 .096
## Within groups (error only)      14611.9 430 33.98          
## 
## 
## ### Post hoc test: Tukey
## 
##                 diff  lwr   upr  p adj
## 30 - 44-18 - 29 -0.99 -2.58 0.59 .307 
## 45+-18 - 29     -1.47 -3.11 0.17 .088 
## 45+-30 - 44     -0.48 -2.1  1.14 .767
one.way <- oneway(sdata$agegp3, y = sdata$tpcoiss, posthoc = 'Tukey') 
#printout a summary of the anova 
one.way
## ### Oneway Anova for y=tpcoiss and x=agegp3 (groups: 18 - 29, 30 - 44, 45+)
## 
## Omega squared: 95% CI = [.01; .08], point estimate = .04
## Eta Squared: 95% CI = [.01; .07], point estimate = .04
## 
##                                       SS  Df     MS    F     p
## Between groups (error + effect)  2565.21   2 1282.6 9.27 <.001
## Within groups (error only)      59057.51 427 138.31           
## 
## 
## ### Post hoc test: Tukey
## 
##                 diff lwr   upr  p adj
## 30 - 44-18 - 29 1.1  -2.11 4.31 .698 
## 45+-18 - 29     5.75 2.44  9.07 <.001
## 45+-30 - 44     4.65 1.36  7.94 .003
one.way <- oneway(sdata$agegp3, y = sdata$tmarlow, posthoc = 'Tukey') 
#printout a summary of the anova 
one.way
## ### Oneway Anova for y=tmarlow and x=agegp3 (groups: 18 - 29, 30 - 44, 45+)
## 
## Omega squared: 95% CI = [.02; .1], point estimate = .05
## Eta Squared: 95% CI = [.03; .09], point estimate = .06
## 
##                                      SS  Df    MS     F     p
## Between groups (error + effect)  104.08   2 52.04 13.18 <.001
## Within groups (error only)      1697.68 430  3.95            
## 
## 
## ### Post hoc test: Tukey
## 
##                 diff lwr   upr  p adj
## 30 - 44-18 - 29 0.82 0.28  1.36 .001 
## 45+-18 - 29     1.18 0.62  1.74 <.001
## 45+-30 - 44     0.36 -0.19 0.92 .278

Build the linear regression models

#Baseline model optimism and social desirability as predictors
model1=lm(sdata$tpcoiss~sdata$toptim+sdata$tmarlow)
stargazer(model1, type="text")
## 
## ===============================================
##                         Dependent variable:    
##                     ---------------------------
##                               tpcoiss          
## -----------------------------------------------
## toptim                       1.321***          
##                               (0.111)          
##                                                
## tmarlow                      1.390***          
##                               (0.240)          
##                                                
## Constant                     23.957***         
##                               (2.655)          
##                                                
## -----------------------------------------------
## Observations                    425            
## R2                             0.319           
## Adjusted R2                    0.315           
## Residual Std. Error      9.936 (df = 422)      
## F Statistic           98.673*** (df = 2; 422)  
## ===============================================
## Note:               *p<0.1; **p<0.05; ***p<0.01
#Check assumptions
#Cooks distance
cooks.distance(model1)
##            1            2            3            4            5 
## 6.502551e-04 3.112103e-03 5.021039e-03 7.497759e-06 9.267960e-04 
##            6            7            8            9           10 
## 2.129260e-02 2.052856e-02 1.188297e-02 1.936683e-02 3.358488e-02 
##           11           12           13           14           15 
## 2.139564e-02 1.556123e-02 4.398459e-08 1.265492e-02 2.582840e-03 
##           16           17           18           19           20 
## 4.616937e-03 5.149222e-03 8.840492e-04 5.011005e-03 8.952237e-04 
##           21           22           23           24           25 
## 5.370918e-04 7.913716e-03 8.967520e-04 6.685494e-04 2.127960e-03 
##           26           27           28           29           31 
## 6.969384e-04 4.119936e-03 1.448822e-04 5.258085e-04 1.213924e-03 
##           32           33           34           35           36 
## 5.812128e-02 2.065601e-04 4.663671e-05 7.460092e-05 1.917659e-03 
##           37           38           39           42           43 
## 5.941958e-04 1.126810e-03 8.840492e-04 2.872576e-04 8.685130e-03 
##           44           46           47           48           49 
## 2.495491e-03 4.142058e-03 1.565434e-03 3.177883e-04 2.223772e-03 
##           50           51           52           53           54 
## 2.437659e-04 8.724413e-06 6.061452e-05 2.950526e-04 4.257250e-03 
##           55           56           57           59           60 
## 7.840810e-03 8.067189e-03 2.847546e-04 1.635709e-04 2.747022e-03 
##           61           62           63           64           65 
## 2.920793e-05 7.320939e-03 1.208383e-03 2.430978e-04 1.935852e-02 
##           66           67           68           69           70 
## 8.151233e-03 2.534092e-02 2.790511e-05 3.471963e-03 5.545158e-04 
##           71           72           73           74           75 
## 3.064099e-03 1.223719e-04 3.052052e-04 4.288045e-04 1.157374e-03 
##           76           77           78           79           80 
## 1.146592e-03 6.574317e-06 6.059503e-03 4.131341e-03 6.094302e-04 
##           81           82           83           84           85 
## 1.861582e-02 2.950526e-04 2.861795e-04 6.228444e-04 2.369347e-03 
##           86           87           88           89           90 
## 9.587631e-04 6.805413e-04 1.008620e-03 9.772701e-05 3.221221e-04 
##           91           92           93           94           95 
## 1.706153e-03 5.989866e-07 6.883234e-03 8.987336e-04 1.643000e-04 
##           96           97           98           99          100 
## 1.401412e-03 4.021454e-03 6.773808e-04 7.071666e-03 8.538847e-05 
##          101          102          103          104          105 
## 6.633850e-05 4.303473e-05 3.432661e-03 3.205774e-05 2.107019e-04 
##          106          107          108          109          110 
## 1.388027e-03 1.334037e-03 2.148214e-05 2.590986e-03 1.159461e-06 
##          111          112          113          114          115 
## 1.915811e-03 1.055125e-02 8.593910e-05 3.606825e-03 5.904843e-05 
##          116          117          118          119          120 
## 3.012951e-05 5.258085e-04 3.086994e-04 8.476463e-03 2.919163e-03 
##          121          122          123          124          125 
## 1.182984e-03 2.909403e-04 1.661236e-03 1.653797e-03 3.549001e-05 
##          126          127          128          129          130 
## 2.078038e-03 4.854594e-03 5.918968e-03 1.758888e-05 1.237878e-04 
##          131          132          133          134          135 
## 1.431355e-04 4.752285e-05 2.245079e-03 1.004769e-04 1.383757e-03 
##          136          137          138          139          140 
## 1.512817e-03 2.223728e-04 4.689399e-03 5.163570e-03 1.636156e-03 
##          141          142          143          144          145 
## 1.653905e-04 1.176677e-04 3.562349e-03 4.054783e-04 3.168772e-04 
##          146          147          148          149          150 
## 3.739763e-05 4.834626e-03 2.730971e-03 1.504892e-04 5.389321e-05 
##          151          152          153          154          155 
## 2.849946e-03 9.859851e-03 4.908827e-03 7.524588e-04 1.000702e-04 
##          156          157          158          159          160 
## 4.056763e-03 3.128635e-05 1.818171e-04 6.867160e-04 4.148999e-04 
##          161          162          163          164          165 
## 3.486215e-03 1.833956e-06 1.341793e-06 3.890415e-04 1.516071e-06 
##          166          167          168          169          170 
## 1.086401e-03 1.386825e-04 3.362393e-04 2.146705e-03 1.169215e-03 
##          171          172          173          174          175 
## 6.293545e-03 9.833695e-05 7.656084e-04 2.045181e-03 1.557336e-04 
##          177          178          179          180          181 
## 1.176677e-04 4.912000e-03 1.372747e-03 4.663926e-03 2.440071e-03 
##          182          183          184          185          186 
## 5.105591e-03 3.332627e-03 4.475102e-07 6.368963e-03 1.129802e-02 
##          187          188          189          190          191 
## 5.618535e-03 6.857586e-03 8.870892e-05 4.529661e-04 1.191831e-03 
##          192          193          194          195          196 
## 1.151615e-03 1.347641e-03 2.624262e-03 3.634924e-06 1.079861e-04 
##          197          198          199          200          201 
## 5.948127e-03 1.112197e-05 2.372697e-05 3.178253e-04 5.271255e-07 
##          202          203          204          205          206 
## 3.113456e-04 3.630448e-04 6.366351e-06 7.214455e-04 5.364698e-05 
##          207          208          209          210          211 
## 1.637006e-04 5.814801e-04 2.003193e-04 7.434163e-04 4.665976e-04 
##          212          213          214          215          216 
## 4.529661e-04 1.912934e-02 6.091161e-06 1.174969e-03 3.539114e-04 
##          217          218          219          220          221 
## 1.790964e-07 2.563216e-03 1.443522e-03 5.311477e-04 6.090387e-03 
##          222          223          224          225          226 
## 1.935501e-04 2.090523e-04 3.372219e-04 2.496893e-03 3.246253e-04 
##          227          228          229          230          231 
## 9.461922e-05 1.033185e-03 2.035038e-03 1.632791e-03 1.727713e-04 
##          232          233          234          235          236 
## 2.373647e-04 1.418739e-05 5.545845e-04 2.420401e-05 3.935649e-03 
##          237          238          239          240          241 
## 3.402395e-03 7.944575e-03 7.434163e-04 3.730773e-06 4.906488e-05 
##          242          243          244          245          246 
## 1.092527e-04 1.522640e-03 2.044247e-08 7.820040e-04 3.991612e-04 
##          247          248          249          250          251 
## 4.265360e-03 1.169998e-03 2.593052e-02 5.204159e-03 5.015126e-03 
##          252          253          254          255          256 
## 2.878781e-03 9.133392e-03 3.900505e-04 1.266901e-03 7.177777e-07 
##          257          258          259          260          261 
## 8.111183e-04 1.659941e-03 7.276857e-05 4.032221e-05 1.090452e-04 
##          262          263          264          265          266 
## 6.152156e-04 3.386712e-04 4.906488e-05 1.768346e-04 3.641208e-04 
##          267          268          269          270          271 
## 8.293862e-05 2.931371e-03 7.562122e-03 1.113867e-03 1.285775e-04 
##          272          273          274          275          276 
## 5.369362e-05 2.221514e-03 4.393732e-03 1.836714e-03 1.634673e-02 
##          277          279          280          281          282 
## 6.571886e-03 2.525762e-05 1.959567e-03 1.031952e-04 1.108088e-03 
##          283          284          285          286          287 
## 8.621910e-03 6.258353e-05 7.033311e-03 2.961425e-03 3.310413e-04 
##          288          289          290          291          292 
## 1.386825e-04 4.275273e-03 9.090313e-06 1.626155e-03 2.061272e-04 
##          293          294          295          296          297 
## 5.398414e-04 7.038954e-04 1.429454e-03 2.808139e-04 1.224816e-03 
##          299          300          301          302          303 
## 1.274288e-03 2.273996e-03 2.670829e-02 1.468360e-03 4.146889e-04 
##          304          305          306          307          308 
## 3.048287e-03 4.545388e-04 9.658637e-06 4.392073e-07 7.495529e-03 
##          309          310          311          312          313 
## 2.537379e-03 1.627368e-03 6.517757e-06 1.993385e-05 2.726138e-03 
##          314          315          316          317          318 
## 4.312277e-04 1.750999e-03 2.343291e-03 2.851173e-03 4.896752e-06 
##          319          320          321          322          323 
## 9.058775e-04 1.718906e-03 1.414605e-03 9.822144e-04 1.569802e-05 
##          324          325          326          327          328 
## 1.013187e-04 1.396138e-05 3.012951e-05 4.506207e-04 5.449837e-04 
##          329          330          331          332          333 
## 1.808162e-03 1.876406e-05 2.909403e-04 8.809182e-05 6.565543e-04 
##          334          335          336          337          338 
## 7.727485e-03 2.837468e-05 1.191831e-03 3.445230e-03 1.675114e-03 
##          339          340          341          343          344 
## 7.785885e-04 9.405542e-04 6.633704e-05 6.535943e-05 1.524739e-04 
##          345          346          347          348          349 
## 4.702745e-05 8.737387e-04 3.838936e-04 1.643000e-04 7.908824e-03 
##          350          351          352          353          354 
## 6.094620e-04 8.032567e-04 2.426273e-03 2.333935e-03 2.071698e-03 
##          355          356          357          358          359 
## 1.708213e-04 2.658016e-03 1.591899e-04 2.593234e-03 1.108950e-04 
##          360          361          362          363          364 
## 3.177883e-04 1.117259e-04 1.822081e-04 6.380893e-04 8.912493e-03 
##          365          366          367          368          369 
## 7.786145e-04 8.311506e-04 9.999980e-05 7.911521e-07 1.493234e-03 
##          370          371          372          373          374 
## 5.435634e-04 6.325072e-03 3.977681e-03 1.097875e-03 1.641530e-03 
##          375          376          377          378          379 
## 1.502801e-02 1.600365e-05 1.049424e-03 1.725883e-04 2.748855e-03 
##          380          381          382          383          384 
## 1.476490e-03 3.220789e-04 8.402395e-06 3.164926e-03 3.142905e-03 
##          385          386          387          388          389 
## 3.635277e-04 1.445644e-03 6.621007e-04 1.095440e-06 1.816590e-04 
##          390          391          392          394          395 
## 3.254265e-03 8.134463e-06 4.347713e-03 7.918642e-04 4.906216e-05 
##          396          397          398          399          400 
## 1.290782e-02 1.547347e-03 4.808448e-04 1.557004e-03 6.464571e-05 
##          401          402          403          404          405 
## 1.250092e-04 7.507327e-05 1.879441e-03 1.004769e-04 1.455795e-03 
##          406          407          408          409          410 
## 3.797437e-06 5.038075e-05 2.543321e-03 2.373799e-03 3.341059e-05 
##          411          412          413          414          415 
## 5.051268e-04 4.228503e-05 7.195078e-05 8.468835e-03 2.917434e-03 
##          416          417          418          419          420 
## 5.706427e-04 7.472142e-07 1.395971e-03 1.052911e-02 1.448435e-03 
##          421          423          424          425          426 
## 4.238825e-05 1.753710e-02 1.241284e-03 3.439611e-04 1.070199e-06 
##          427          428          429          430          431 
## 1.040129e-04 3.523737e-04 3.010825e-04 4.666765e-04 7.559538e-04 
##          432          433          435          437          439 
## 4.752285e-05 4.500997e-03 2.245644e-03 4.560286e-05 4.375925e-03
#Plot Cooks distance
plot(cooks.distance(model1), ylab="Cook's statistic")

#Create histogram
#A density plot of the residuals
plot(density(resid(model1))) 

#Create a QQ plotqqPlot(model, main="QQ Plot") #qq plot for studentized resid 
leveragePlots(model1) # leverage plots

#Collinearity
vifmodel<-vif(model1)
#Calculate tolerance
1/vifmodel
##  sdata$toptim sdata$tmarlow 
##     0.9836225     0.9836225
#Second model adding in stress
model2=lm(sdata$tpcoiss~sdata$toptim+sdata$tmarlow+sdata$tpstress)
summary(model2)
## 
## Call:
## lm(formula = sdata$tpcoiss ~ sdata$toptim + sdata$tmarlow + sdata$tpstress)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -34.253  -5.034   0.271   5.701  28.070 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    58.55471    4.40587  13.290  < 2e-16 ***
## sdata$toptim    0.84267    0.11264   7.481 4.38e-13 ***
## sdata$tmarlow   0.98044    0.22400   4.377 1.52e-05 ***
## sdata$tpstress -0.81767    0.08663  -9.438  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.019 on 418 degrees of freedom
##   (17 observations deleted due to missingness)
## Multiple R-squared:  0.4402, Adjusted R-squared:  0.4361 
## F-statistic: 109.5 on 3 and 418 DF,  p-value: < 2.2e-16
#Check assumptions
#Cooks distance
cooks.distance(model2)
##            1            2            3            4            5 
## 4.386133e-04 1.122128e-02 1.841839e-02 8.104241e-04 1.677983e-03 
##            6            7            8            9           10 
## 3.951943e-02 1.278646e-02 1.858489e-02 1.124209e-02 1.319163e-02 
##           11           12           13           14           15 
## 2.095067e-02 3.536274e-02 2.078877e-04 1.510310e-02 4.006136e-04 
##           16           17           18           19           20 
## 2.818643e-03 5.081900e-03 2.784382e-02 6.187649e-04 9.385656e-04 
##           21           22           23           24           25 
## 1.692352e-04 2.913993e-02 2.425825e-03 1.963788e-03 1.334735e-03 
##           26           27           28           29           31 
## 2.386220e-05 5.446850e-03 1.513688e-03 9.679650e-07 3.836379e-04 
##           32           33           34           35           36 
## 3.596554e-02 6.542349e-04 1.963583e-04 4.635160e-04 1.006445e-03 
##           37           38           39           42           43 
## 3.779544e-04 6.207859e-04 2.893225e-03 1.518792e-03 6.252466e-03 
##           44           46           47           48           49 
## 4.803713e-04 3.623659e-03 1.038313e-02 2.437726e-03 5.997050e-04 
##           50           51           52           53           54 
## 6.230510e-03 1.268997e-04 1.895822e-03 1.023493e-03 5.358517e-03 
##           55           56           57           59           60 
## 6.697990e-02 4.617651e-02 7.650889e-04 4.571700e-03 3.066488e-03 
##           61           62           63           64           65 
## 3.898795e-04 6.702801e-03 9.716547e-04 8.870216e-05 7.531584e-03 
##           66           67           68           69           70 
## 1.070167e-02 1.516978e-02 5.397230e-05 4.659988e-04 8.643770e-03 
##           71           72           73           74           75 
## 1.326955e-03 4.032575e-03 7.038843e-04 6.651905e-04 3.143702e-04 
##           76           77           78           79           80 
## 4.152421e-04 1.621811e-05 7.732975e-04 1.689876e-02 1.567265e-05 
##           81           82           83           84           85 
## 1.402302e-02 6.050244e-04 3.492398e-04 5.618475e-04 1.600254e-03 
##           86           87           88           89           90 
## 3.831070e-06 3.579970e-04 3.061522e-04 1.108126e-03 1.156157e-03 
##           91           92           93           94           95 
## 1.376175e-03 3.209951e-04 3.646189e-03 2.361536e-04 3.462976e-04 
##           96           97           98           99          100 
## 8.202307e-05 1.680604e-04 7.306675e-04 4.747362e-03 7.856356e-05 
##          101          102          103          104          105 
## 1.008669e-05 6.809411e-04 1.238707e-02 2.383199e-04 6.062445e-04 
##          106          107          108          109          110 
## 1.233405e-03 8.383219e-04 2.944082e-04 4.329760e-04 1.011100e-03 
##          111          112          113          114          115 
## 1.938113e-03 7.450220e-03 1.025415e-04 2.037185e-03 1.223689e-04 
##          116          117          118          119          120 
## 5.507892e-06 3.817765e-04 3.338964e-05 7.695163e-03 2.624638e-03 
##          121          122          123          124          125 
## 9.518047e-04 6.441381e-05 1.690502e-03 1.098883e-03 6.995576e-04 
##          126          127          128          129          130 
## 5.990789e-03 1.280055e-02 3.799758e-03 6.759402e-05 6.630600e-05 
##          131          132          133          134          135 
## 2.479354e-04 3.051437e-05 5.508596e-04 1.585305e-04 1.050701e-03 
##          136          137          138          139          140 
## 7.095943e-04 8.935817e-04 6.350803e-04 4.076625e-02 1.429977e-02 
##          141          142          143          144          145 
## 1.206158e-03 5.763347e-04 2.135666e-03 8.705416e-09 1.884086e-03 
##          146          147          148          149          150 
## 4.320267e-05 3.078351e-03 2.194644e-03 1.164725e-05 6.621026e-05 
##          151          152          153          154          155 
## 1.402424e-02 6.955861e-03 3.331298e-03 1.002329e-03 1.530437e-04 
##          156          157          158          159          160 
## 2.578946e-03 7.411840e-06 4.302432e-04 3.132903e-03 5.284563e-05 
##          161          162          163          164          165 
## 1.908219e-03 1.067378e-03 5.755838e-04 6.092257e-04 5.196341e-05 
##          166          167          168          169          170 
## 8.455585e-04 3.543527e-04 1.318425e-04 1.881276e-03 9.157110e-04 
##          171          172          173          174          175 
## 6.946290e-03 5.300279e-05 2.127500e-05 1.295259e-03 7.237538e-05 
##          177          178          179          180          181 
## 8.855640e-05 4.740295e-03 1.064043e-03 3.319243e-03 3.817172e-04 
##          182          183          184          185          186 
## 4.436219e-03 2.669174e-03 9.347653e-04 6.462383e-03 1.053434e-02 
##          187          188          189          190          191 
## 5.679416e-03 4.930282e-03 4.027091e-04 4.504617e-04 1.643329e-03 
##          192          193          194          195          196 
## 4.818361e-05 1.592021e-03 2.571586e-03 4.890655e-05 2.127256e-03 
##          197          198          199          200          201 
## 6.103995e-03 1.201321e-04 1.864803e-04 6.200510e-05 1.986242e-04 
##          202          203          204          205          206 
## 2.126468e-04 1.579025e-05 2.576602e-05 2.475584e-04 1.012185e-04 
##          207          208          209          210          211 
## 8.324393e-05 1.107312e-03 3.054095e-07 2.217814e-04 4.681305e-04 
##          212          213          214          215          216 
## 1.115400e-03 1.223851e-02 5.757810e-06 4.472341e-04 8.394730e-04 
##          217          218          219          220          221 
## 9.312336e-06 3.066385e-03 4.856027e-04 5.113950e-04 4.688919e-03 
##          222          223          224          225          226 
## 8.904989e-04 2.971711e-04 1.265362e-04 2.381366e-03 4.750834e-06 
##          227          228          229          230          231 
## 2.481636e-04 1.304818e-04 1.020656e-03 1.199490e-02 2.910689e-04 
##          232          233          234          235          236 
## 1.204419e-04 7.463380e-06 7.239221e-04 1.370971e-05 3.586538e-03 
##          237          238          239          240          241 
## 1.352064e-03 7.742845e-03 3.192741e-05 9.414624e-06 1.400134e-05 
##          242          243          244          245          246 
## 2.247768e-04 2.751394e-03 4.638816e-06 1.092434e-03 4.336220e-05 
##          247          248          249          250          251 
## 5.012996e-03 1.846043e-03 2.276583e-02 3.502830e-03 3.373733e-03 
##          252          253          254          255          256 
## 1.637806e-03 7.437340e-03 6.446441e-05 1.249926e-03 1.628093e-05 
##          257          258          259          260          261 
## 2.089036e-03 8.269490e-04 6.272452e-04 1.555396e-04 5.870607e-05 
##          262          263          264          265          266 
## 1.277570e-03 8.920941e-05 1.653760e-03 5.820557e-06 2.716542e-04 
##          267          268          269          270          271 
## 2.781698e-05 2.372743e-03 7.082099e-03 9.568469e-04 5.097000e-06 
##          272          273          274          275          276 
## 2.253401e-04 1.783613e-03 3.420871e-03 1.023987e-03 9.543107e-03 
##          277          279          280          281          282 
## 8.586335e-03 1.260799e-05 4.736527e-03 1.722811e-07 5.328438e-05 
##          283          284          285          286          287 
## 9.064871e-03 2.146134e-05 5.317210e-03 1.760873e-03 1.905966e-05 
##          288          289          290          291          292 
## 4.640265e-05 4.380822e-03 1.093820e-04 9.877358e-04 1.344988e-04 
##          293          294          295          296          297 
## 1.585704e-04 3.916155e-04 7.051611e-04 1.610758e-04 4.548479e-04 
##          299          300          301          302          303 
## 4.538674e-04 1.201836e-03 2.143479e-02 1.205541e-03 3.031590e-04 
##          304          305          306          307          308 
## 8.962552e-04 1.114583e-04 1.393493e-04 5.159693e-06 6.983783e-03 
##          309          310          311          312          313 
## 2.251318e-03 2.902743e-03 2.288025e-04 6.227432e-04 2.346624e-03 
##          314          315          316          317          318 
## 4.089628e-05 1.387249e-03 2.760761e-03 2.495428e-03 3.530648e-05 
##          319          320          321          322          323 
## 3.380011e-04 5.835935e-04 2.496055e-03 2.774031e-04 2.308443e-04 
##          324          325          326          327          328 
## 1.686315e-04 1.143373e-03 9.677548e-05 1.683532e-03 7.132650e-04 
##          329          330          331          332          333 
## 1.353526e-03 5.250248e-06 1.256242e-04 7.959785e-05 4.593538e-04 
##          334          335          336          337          338 
## 5.517579e-03 1.932397e-05 5.013628e-04 3.298492e-03 6.990907e-04 
##          339          340          341          343          344 
## 4.655428e-05 3.522176e-04 2.364236e-05 2.207084e-05 3.881046e-05 
##          345          346          347          348          349 
## 1.627580e-05 1.514174e-03 1.055266e-04 1.433574e-04 6.123495e-03 
##          350          351          352          353          354 
## 2.574264e-04 3.304918e-04 6.589807e-04 8.551661e-04 1.323455e-03 
##          355          356          357          358          359 
## 5.606057e-06 2.238127e-03 5.255332e-05 1.280023e-03 1.999500e-05 
##          360          361          362          363          364 
## 5.766521e-05 6.914136e-05 6.995432e-05 6.633236e-04 2.361188e-02 
##          365          366          367          368          369 
## 4.722831e-04 5.031333e-03 2.895058e-04 1.072320e-04 1.141158e-03 
##          370          371          372          373          374 
## 4.935098e-04 4.322763e-03 4.292238e-03 5.731036e-04 1.015266e-03 
##          375          376          377          378          379 
## 1.222428e-02 4.520365e-04 8.836257e-04 4.301008e-06 1.025994e-03 
##          380          381          382          383          384 
## 1.155738e-03 7.250348e-05 6.368504e-04 2.767466e-03 2.247564e-03 
##          385          386          387          388          389 
## 1.409342e-04 8.629613e-05 3.288017e-06 1.437695e-05 1.175090e-06 
##          390          391          392          394          395 
## 2.532978e-03 7.736559e-05 4.522563e-03 9.199994e-04 1.636145e-09 
##          396          397          398          399          400 
## 9.179859e-03 1.939028e-03 1.048373e-03 2.006180e-03 9.043328e-05 
##          401          402          403          404          405 
## 2.043244e-04 1.819969e-04 5.282918e-03 1.047493e-05 3.865099e-04 
##          406          407          408          409          410 
## 1.595736e-04 9.715567e-07 1.649315e-03 4.660351e-04 2.702553e-06 
##          411          412          413          414          415 
## 3.265448e-04 2.975850e-05 1.782046e-04 6.869838e-03 1.578350e-03 
##          416          417          418          419          420 
## 1.320362e-03 6.239612e-05 3.516138e-03 6.107803e-03 5.268216e-03 
##          421          423          424          425          426 
## 8.229372e-06 1.461833e-02 9.274811e-04 1.289364e-05 2.072355e-05 
##          427          428          429          430          431 
## 2.455461e-05 3.639891e-04 2.562616e-04 2.381704e-04 5.211356e-04 
##          432          437 
## 3.704604e-04 1.143229e-03
#Create histogram
 
#A density plot of the residuals
plot(density(resid(model2))) 

#Create a QQ plot qPlot(model, main="QQ Plot") #qq plot for studentized resid 
leveragePlots(model2) # leverage plots

#Collinearity
vif(model2)
##   sdata$toptim  sdata$tmarlow sdata$tpstress 
##       1.279007       1.057358       1.330851
sqrt(vif(model2)) 
##   sdata$toptim  sdata$tmarlow sdata$tpstress 
##       1.130932       1.028279       1.153625
#Model 3 adding in gender 
#dummycode
sdata$gender=recode(sdata$sex,'0=1;1=2')
model3=lm(sdata$tpcoiss~sdata$toptim+sdata$tmarlow+sdata$tpstress+sdata$sex)
summary(model3)
## 
## Call:
## lm(formula = sdata$tpcoiss ~ sdata$toptim + sdata$tmarlow + sdata$tpstress + 
##     sdata$sex)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -34.785  -5.241   0.139   5.801  28.424 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    56.17192    4.56967  12.292  < 2e-16 ***
## sdata$toptim    0.86391    0.11285   7.655 1.36e-13 ***
## sdata$tmarlow   1.03612    0.22525   4.600 5.62e-06 ***
## sdata$tpstress -0.78439    0.08814  -8.899  < 2e-16 ***
## sdata$sexMALES  1.71591    0.90775   1.890   0.0594 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.991 on 417 degrees of freedom
##   (17 observations deleted due to missingness)
## Multiple R-squared:  0.4449, Adjusted R-squared:  0.4396 
## F-statistic: 83.56 on 4 and 417 DF,  p-value: < 2.2e-16
plot(model3)

#Check assumptions
#Cooks distance
cooks.distance(model3)
##            1            2            3            4            5 
## 4.197679e-04 1.211857e-02 1.618401e-02 5.551629e-04 2.264980e-03 
##            6            7            8            9           10 
## 3.231304e-02 9.922411e-03 2.344161e-02 8.680495e-03 1.496076e-02 
##           11           12           13           14           15 
## 2.101761e-02 3.420758e-02 9.772502e-05 1.183614e-02 2.118482e-04 
##           16           17           18           19           20 
## 2.169598e-03 3.911430e-03 2.351318e-02 3.793327e-04 8.606866e-04 
##           21           22           23           24           25 
## 2.142289e-05 2.556939e-02 2.569283e-03 2.011169e-03 1.843008e-03 
##           26           27           28           29           31 
## 3.790730e-06 4.373871e-03 1.705208e-03 6.581347e-06 8.048003e-04 
##           32           33           34           35           36 
## 2.820298e-02 7.733624e-04 8.887682e-05 3.814116e-04 1.321204e-03 
##           37           38           39           42           43 
## 3.742400e-04 8.454544e-04 3.412412e-03 9.610690e-04 4.561408e-03 
##           44           46           47           48           49 
## 3.149815e-04 2.948345e-03 9.000356e-03 1.945060e-03 3.657561e-04 
##           50           51           52           53           54 
## 5.664595e-03 2.798421e-04 1.735660e-03 6.136839e-04 5.376345e-03 
##           55           56           57           59           60 
## 5.875351e-02 4.307659e-02 8.650980e-04 4.020425e-03 2.287152e-03 
##           61           62           63           64           65 
## 2.855970e-04 6.774712e-03 9.272567e-04 1.290323e-04 5.768160e-03 
##           66           67           68           69           70 
## 8.400061e-03 1.183437e-02 3.194868e-05 7.653155e-04 7.403000e-03 
##           71           72           73           74           75 
## 9.829533e-04 3.375666e-03 9.262198e-04 6.846940e-04 7.718024e-04 
##           76           77           78           79           80 
## 2.564665e-04 4.014454e-07 6.411572e-04 1.537949e-02 1.245155e-06 
##           81           82           83           84           85 
## 1.015045e-02 7.780475e-04 3.362903e-04 8.177173e-04 2.374995e-03 
##           86           87           88           89           90 
## 3.319909e-05 7.542612e-04 1.770498e-04 1.066118e-03 1.332752e-03 
##           91           92           93           94           95 
## 1.432210e-03 1.959148e-04 4.195844e-03 6.321568e-04 5.037242e-04 
##           96           97           98           99          100 
## 2.854249e-05 2.739247e-05 8.179369e-04 7.255416e-03 1.756023e-04 
##          101          102          103          104          105 
## 3.001939e-05 7.345742e-04 1.164994e-02 4.180816e-04 8.294681e-04 
##          106          107          108          109          110 
## 1.951387e-03 1.144373e-03 4.222578e-04 2.941887e-04 1.290234e-03 
##          111          112          113          114          115 
## 2.272432e-03 8.906600e-03 3.313974e-05 2.603343e-03 1.867603e-04 
##          116          117          118          119          120 
## 6.797506e-06 6.601685e-04 1.803555e-07 6.741639e-03 3.422203e-03 
##          121          122          123          124          125 
## 1.503357e-03 1.788088e-05 2.180131e-03 1.285808e-03 9.625718e-04 
##          126          127          128          129          130 
## 4.953154e-03 1.233853e-02 2.830108e-03 1.493332e-05 1.607792e-06 
##          131          132          133          134          135 
## 1.364789e-04 1.130918e-04 3.546110e-04 1.718545e-04 1.339119e-03 
##          136          137          138          139          140 
## 1.268833e-03 1.004576e-03 3.598073e-04 3.457427e-02 1.177440e-02 
##          141          142          143          144          145 
## 1.465423e-03 6.586219e-04 1.692150e-03 5.728757e-06 1.557841e-03 
##          146          147          148          149          150 
## 1.118464e-05 3.838866e-03 3.106583e-03 8.568775e-05 3.553826e-05 
##          151          152          153          154          155 
## 1.246649e-02 8.086788e-03 2.643602e-03 1.456764e-03 3.460046e-04 
##          156          157          158          159          160 
## 1.996944e-03 1.239686e-06 5.470631e-04 2.295693e-03 2.132351e-05 
##          161          162          163          164          165 
## 1.518723e-03 7.688412e-04 3.637548e-04 4.600586e-04 1.630315e-05 
##          166          167          168          169          170 
## 9.387885e-04 3.109898e-04 2.811556e-05 2.696406e-03 1.006388e-03 
##          171          172          173          174          175 
## 6.305138e-03 1.826157e-05 1.139847e-04 9.890679e-04 4.622274e-05 
##          177          178          179          180          181 
## 5.052888e-05 4.211214e-03 2.027140e-03 2.798842e-03 6.994851e-04 
##          182          183          184          185          186 
## 6.132901e-03 3.532210e-03 6.137388e-04 7.337603e-03 8.815952e-03 
##          187          188          189          190          191 
## 5.383469e-03 5.982091e-03 2.209876e-04 5.225183e-04 1.736916e-03 
##          192          193          194          195          196 
## 1.868019e-04 1.227234e-03 2.975273e-03 9.202072e-05 2.526248e-03 
##          197          198          199          200          201 
## 4.980183e-03 1.872972e-04 1.115488e-04 1.887004e-04 7.764134e-05 
##          202          203          204          205          206 
## 4.584208e-04 7.579627e-05 5.554306e-06 1.657696e-04 1.571302e-04 
##          207          208          209          210          211 
## 1.887963e-04 1.178353e-03 8.196134e-06 1.292070e-04 6.988422e-04 
##          212          213          214          215          216 
## 1.836346e-03 9.680570e-03 3.959528e-07 8.244072e-04 8.892333e-04 
##          217          218          219          220          221 
## 8.193632e-07 3.196681e-03 3.262213e-04 6.803670e-04 3.960848e-03 
##          222          223          224          225          226 
## 7.875614e-04 4.034885e-04 9.144393e-05 2.000980e-03 1.580416e-06 
##          227          228          229          230          231 
## 1.251735e-04 4.853038e-05 8.364939e-04 1.050282e-02 4.277987e-04 
##          232          233          234          235          236 
## 3.209268e-04 2.685046e-05 1.229495e-03 1.239696e-06 3.214409e-03 
##          237          238          239          240          241 
## 9.698840e-04 1.182820e-02 1.021546e-04 5.277873e-07 1.127162e-04 
##          242          243          244          245          246 
## 1.881968e-04 2.164420e-03 4.245212e-07 1.509754e-03 1.964100e-04 
##          247          248          249          250          251 
## 6.573517e-03 1.950828e-03 2.166570e-02 3.045035e-03 3.120855e-03 
##          252          253          254          255          256 
## 1.417851e-03 9.999129e-03 2.069329e-05 2.226214e-03 4.527933e-06 
##          257          258          259          260          261 
## 3.130802e-03 6.585050e-04 6.513967e-04 1.986348e-04 4.878517e-06 
##          262          263          264          265          266 
## 1.114737e-03 2.094699e-05 1.568792e-03 1.800409e-05 4.649115e-04 
##          267          268          269          270          271 
## 1.585284e-06 1.964130e-03 5.910648e-03 1.432358e-03 2.332845e-08 
##          272          273          274          275          276 
## 4.030685e-04 2.754178e-03 3.626445e-03 9.083886e-04 1.011470e-02 
##          277          279          280          281          282 
## 8.473261e-03 3.585344e-05 4.637712e-03 9.694950e-06 1.749809e-04 
##          283          284          285          286          287 
## 8.280766e-03 6.874586e-05 4.168752e-03 2.036204e-03 7.963922e-05 
##          288          289          290          291          292 
## 3.215642e-06 3.875906e-03 1.643751e-04 9.433776e-04 8.932770e-05 
##          293          294          295          296          297 
## 9.516064e-05 2.844443e-04 4.194760e-04 1.148010e-04 7.885892e-04 
##          299          300          301          302          303 
## 1.053573e-03 2.040621e-03 1.746390e-02 1.459593e-03 6.959472e-04 
##          304          305          306          307          308 
## 5.881585e-04 2.134193e-04 2.817285e-04 6.740716e-05 6.774610e-03 
##          309          310          311          312          313 
## 3.136208e-03 2.552473e-03 1.167534e-04 5.114699e-04 2.411268e-03 
##          314          315          316          317          318 
## 1.010636e-04 1.428834e-03 2.352240e-03 3.147816e-03 5.248036e-06 
##          319          320          321          322          323 
## 1.550405e-04 7.489822e-04 2.098045e-03 5.707228e-04 4.662413e-04 
##          324          325          326          327          328 
## 2.157255e-04 1.327133e-03 1.356481e-04 1.638573e-03 5.604195e-04 
##          329          330          331          332          333 
## 1.285783e-03 5.643174e-07 2.122820e-04 1.265711e-04 4.761796e-04 
##          334          335          336          337          338 
## 5.093441e-03 1.105760e-06 3.573770e-04 4.116176e-03 4.816687e-04 
##          339          340          341          343          344 
## 5.366328e-06 5.590812e-04 1.630920e-06 2.778076e-06 1.104570e-04 
##          345          346          347          348          349 
## 1.016314e-04 1.449896e-03 5.954211e-05 1.012711e-04 4.983936e-03 
##          350          351          352          353          354 
## 1.939200e-04 2.196996e-04 1.171223e-03 5.847426e-04 1.630944e-03 
##          355          356          357          358          359 
## 8.677706e-08 2.287167e-03 2.132634e-05 2.309821e-03 2.387297e-06 
##          360          361          362          363          364 
## 1.733529e-04 4.974783e-05 2.299280e-04 8.510232e-04 2.006994e-02 
##          365          366          367          368          369 
## 4.177580e-04 5.147529e-03 5.117771e-04 2.998917e-04 1.351073e-03 
##          370          371          372          373          374 
## 6.899045e-04 5.228499e-03 4.600990e-03 4.669527e-04 8.438978e-04 
##          375          376          377          378          379 
## 1.628354e-02 7.624859e-04 8.018057e-04 4.777453e-06 1.873800e-03 
##          380          381          382          383          384 
## 1.819473e-03 1.832969e-04 4.039127e-04 3.244130e-03 1.685125e-03 
##          385          386          387          388          389 
## 1.025939e-04 3.137450e-05 2.509090e-05 9.521722e-05 3.301360e-06 
##          390          391          392          394          395 
## 2.001649e-03 1.905834e-05 6.323092e-03 1.292753e-03 8.664254e-06 
##          396          397          398          399          400 
## 7.407007e-03 2.359656e-03 1.803314e-03 2.069562e-03 4.253547e-05 
##          401          402          403          404          405 
## 1.136323e-04 1.140534e-04 4.975520e-03 4.496540e-05 2.558134e-04 
##          406          407          408          409          410 
## 1.133500e-04 1.842323e-05 2.351419e-03 8.634116e-04 2.216123e-05 
##          411          412          413          414          415 
## 4.282417e-04 6.879309e-06 4.223486e-04 5.942391e-03 1.266087e-03 
##          416          417          418          419          420 
## 1.072475e-03 9.572363e-06 3.647850e-03 4.771934e-03 4.535670e-03 
##          421          423          424          425          426 
## 3.759026e-05 1.471750e-02 1.470953e-03 3.668430e-05 5.600332e-05 
##          427          428          429          430          431 
## 9.061783e-05 3.107488e-04 1.358236e-04 4.546927e-04 6.059712e-04 
##          432          437 
## 5.939650e-04 1.150430e-03
#Create histogram
 
#A density plot of the residuals
plot(density(resid(model3))) 

#Create a QQ plot qqPlot(model, main="QQ Plot") #qq plot for studentized resid 
leveragePlots(model3) # leverage plots

#Collinearity
vif(model3)
##   sdata$toptim  sdata$tmarlow sdata$tpstress      sdata$sex 
##       1.291821       1.075754       1.386133       1.050668
sqrt(vif(model3)) 
##   sdata$toptim  sdata$tmarlow sdata$tpstress      sdata$sex 
##       1.136583       1.037185       1.177342       1.025021
#Model 4 adding in interaction term gender*stress
#create interaction term
sdata$intgenstress=as.numeric(sdata$gender)*sdata$tpstress
model4=lm(sdata$tpcoiss~sdata$toptim+sdata$tmarlow+sdata$tpstress+sdata$gender+sdata$intgenstress)
summary(model4)
## 
## Call:
## lm(formula = sdata$tpcoiss ~ sdata$toptim + sdata$tmarlow + sdata$tpstress + 
##     sdata$gender + sdata$intgenstress)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -35.275  -5.186   0.235   5.836  28.198 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        55.48477    4.86777  11.398  < 2e-16 ***
## sdata$toptim        0.86682    0.11318   7.659 1.33e-13 ***
## sdata$tmarlow       1.02637    0.22671   4.527 7.81e-06 ***
## sdata$tpstress     -0.69471    0.23450  -2.962  0.00323 ** 
## sdata$genderMALES   3.42931    4.24948   0.807  0.42013    
## sdata$intgenstress -0.06507    0.15766  -0.413  0.68000    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9 on 416 degrees of freedom
##   (17 observations deleted due to missingness)
## Multiple R-squared:  0.4451, Adjusted R-squared:  0.4385 
## F-statistic: 66.75 on 5 and 416 DF,  p-value: < 2.2e-16
plot(model4)

#Check assumptions
#Cooks distance
cooks.distance(model4)
##            1            2            3            4            5 
## 3.683116e-04 1.178676e-02 1.644426e-02 5.921484e-04 1.911109e-03 
##            6            7            8            9           10 
## 3.002889e-02 1.058362e-02 3.269252e-02 1.089707e-02 2.645437e-02 
##           11           12           13           14           15 
## 1.950181e-02 3.899282e-02 2.062466e-04 1.560389e-02 1.749112e-04 
##           16           17           18           19           20 
## 2.178498e-03 4.778911e-03 2.797210e-02 6.712949e-04 7.054063e-04 
##           21           22           23           24           25 
## 1.207878e-04 2.932034e-02 2.164039e-03 1.732762e-03 1.776822e-03 
##           26           27           28           29           31 
## 8.438375e-06 5.174328e-03 1.477976e-03 1.689095e-05 6.693014e-04 
##           32           33           34           35           36 
## 2.870488e-02 1.367956e-03 7.293801e-05 3.527067e-04 1.119045e-03 
##           37           38           39           42           43 
## 3.316852e-04 7.061671e-04 4.842655e-03 1.009167e-03 5.502894e-03 
##           44           46           47           48           49 
## 3.682338e-04 3.127571e-03 8.421178e-03 3.146750e-03 4.488304e-04 
##           50           51           52           53           54 
## 6.177542e-03 2.151253e-04 1.680959e-03 6.133201e-04 4.631203e-03 
##           55           56           57           59           60 
## 1.001518e-01 7.167624e-02 7.277119e-04 3.383246e-03 2.016119e-03 
##           61           62           63           64           65 
## 2.350115e-04 5.663261e-03 7.809555e-04 1.041914e-04 5.772658e-03 
##           66           67           68           69           70 
## 8.643961e-03 1.290567e-02 3.897609e-05 1.109233e-03 6.436888e-03 
##           71           72           73           74           75 
## 8.623124e-04 3.626219e-03 7.881128e-04 8.903274e-04 7.153656e-04 
##           76           77           78           79           80 
## 2.233232e-04 2.984345e-08 4.815933e-04 1.510601e-02 5.630290e-06 
##           81           82           83           84           85 
## 1.047456e-02 1.711374e-03 2.787357e-04 7.954411e-04 2.579801e-03 
##           86           87           88           89           90 
## 1.089499e-04 7.386730e-04 1.983170e-04 1.420885e-03 1.129245e-03 
##           91           92           93           94           95 
## 1.195295e-03 2.300407e-04 5.443813e-03 6.965125e-04 9.945682e-04 
##           96           97           98           99          100 
## 4.931185e-05 5.330733e-06 7.924720e-04 8.064176e-03 2.843402e-04 
##          101          102          103          104          105 
## 2.403436e-05 7.286398e-04 1.084999e-02 4.658209e-04 7.429898e-04 
##          106          107          108          109          110 
## 1.828286e-03 1.203475e-03 3.675654e-04 3.681127e-04 1.756676e-03 
##          111          112          113          114          115 
## 1.981103e-03 9.754931e-03 2.354276e-05 3.355610e-03 1.756504e-04 
##          116          117          118          119          120 
## 3.399336e-06 5.491710e-04 6.431870e-06 9.923681e-03 3.236055e-03 
##          121          122          123          124          125 
## 1.624381e-03 1.172468e-05 2.206231e-03 1.337213e-03 8.058774e-04 
##          126          127          128          129          130 
## 4.108480e-03 1.170423e-02 3.615455e-03 8.760133e-06 1.472871e-06 
##          131          132          133          134          135 
## 1.144466e-04 9.985317e-05 4.278511e-04 1.536422e-04 1.317295e-03 
##          136          137          138          139          140 
## 1.372503e-03 9.001797e-04 3.873307e-04 3.479622e-02 1.093127e-02 
##          141          142          143          144          145 
## 1.703842e-03 5.474786e-04 1.924143e-03 3.277167e-06 2.786034e-03 
##          146          147          148          149          150 
## 8.920424e-06 4.944636e-03 2.807581e-03 1.324993e-04 4.125021e-05 
##          151          152          153          154          155 
## 1.116912e-02 6.872314e-03 2.655669e-03 1.348741e-03 5.622060e-04 
##          156          157          158          159          160 
## 1.801882e-03 4.300273e-06 5.199456e-04 2.348412e-03 2.588558e-05 
##          161          162          163          164          165 
## 1.433510e-03 7.745445e-04 3.306319e-04 4.034340e-04 1.404673e-05 
##          166          167          168          169          170 
## 7.893235e-04 2.621461e-04 1.683682e-05 2.722357e-03 8.687618e-04 
##          171          172          173          174          175 
## 6.523981e-03 1.382915e-05 1.082414e-04 1.165182e-03 3.988461e-05 
##          177          178          179          180          181 
## 4.140182e-05 5.271212e-03 2.711268e-03 3.018363e-03 7.328011e-04 
##          182          183          184          185          186 
## 5.153066e-03 3.042363e-03 5.065314e-04 6.485450e-03 1.062737e-02 
##          187          188          189          190          191 
## 4.907789e-03 5.088367e-03 4.011871e-04 4.368638e-04 1.502905e-03 
##          192          193          194          195          196 
## 2.073765e-04 1.662333e-03 2.553593e-03 7.616832e-05 2.195589e-03 
##          197          198          199          200          201 
## 4.695773e-03 1.739894e-04 1.441452e-04 1.828951e-04 5.316878e-05 
##          202          203          204          205          206 
## 4.573369e-04 1.054271e-04 8.035693e-06 1.711428e-04 1.323312e-04 
##          207          208          209          210          211 
## 1.627571e-04 1.017564e-03 1.343025e-05 1.048791e-04 5.848871e-04 
##          212          213          214          215          216 
## 1.798831e-03 8.274625e-03 7.276398e-07 6.814074e-04 7.999788e-04 
##          217          218          219          220          221 
## 2.636695e-06 4.196595e-03 3.326340e-04 5.668468e-04 3.560563e-03 
##          222          223          224          225          226 
## 7.932861e-04 5.116409e-04 8.555473e-05 1.993805e-03 2.969881e-06 
##          227          228          229          230          231 
## 1.038617e-04 3.587097e-05 7.723367e-04 1.043787e-02 3.632631e-04 
##          232          233          234          235          236 
## 2.918906e-04 2.170598e-05 1.274902e-03 1.191147e-06 2.706881e-03 
##          237          238          239          240          241 
## 1.002980e-03 1.127884e-02 1.462014e-04 2.341247e-06 9.691864e-05 
##          242          243          244          245          246 
## 1.562080e-04 1.918112e-03 1.208516e-06 1.577807e-03 2.246451e-04 
##          247          248          249          250          251 
## 6.856845e-03 1.643265e-03 1.836275e-02 3.000762e-03 2.938401e-03 
##          252          253          254          255          256 
## 1.424394e-03 1.059559e-02 1.504406e-05 1.875266e-03 3.270413e-06 
##          257          258          259          260          261 
## 3.020612e-03 5.479769e-04 5.347724e-04 1.788675e-04 7.067421e-06 
##          262          263          264          265          266 
## 1.579329e-03 1.297818e-05 1.306728e-03 1.618743e-05 4.197649e-04 
##          267          268          269          270          271 
## 5.603899e-07 1.682458e-03 5.078914e-03 1.249098e-03 2.138688e-07 
##          272          273          274          275          276 
## 3.637804e-04 2.297152e-03 3.625006e-03 9.706161e-04 8.453090e-03 
##          277          279          280          281          282 
## 8.221500e-03 2.903182e-05 3.917443e-03 1.289357e-05 1.876943e-04 
##          283          284          285          286          287 
## 7.811538e-03 6.241586e-05 3.669516e-03 2.260326e-03 9.696378e-05 
##          288          289          290          291          292 
## 1.075501e-05 3.439622e-03 1.476867e-04 9.451931e-04 8.043664e-05 
##          293          294          295          296          297 
## 8.006605e-05 2.608237e-04 3.642120e-04 1.110590e-04 6.912347e-04 
##          299          300          301          302          303 
## 1.037926e-03 2.302189e-03 1.506446e-02 1.213464e-03 5.784579e-04 
##          304          305          306          307          308 
## 5.477090e-04 2.494791e-04 2.621553e-04 5.609695e-05 6.037795e-03 
##          309          310          311          312          313 
## 2.623062e-03 2.190796e-03 1.184799e-04 4.595411e-04 2.136557e-03 
##          314          315          316          317          318 
## 1.178762e-04 1.255140e-03 1.957706e-03 2.617487e-03 4.647854e-06 
##          319          320          321          322          323 
## 1.203079e-04 7.784063e-04 1.996550e-03 5.339854e-04 4.768896e-04 
##          324          325          326          327          328 
## 1.873916e-04 1.377090e-03 1.124370e-04 1.388773e-03 7.133237e-04 
##          329          330          331          332          333 
## 1.068561e-03 6.688153e-07 2.277190e-04 1.005425e-04 4.126710e-04 
##          334          335          336          337          338 
## 4.239403e-03 3.872964e-07 3.208166e-04 3.558723e-03 3.987121e-04 
##          339          340          341          343          344 
## 4.336547e-07 6.265775e-04 6.362877e-07 2.184722e-06 1.503205e-04 
##          345          346          347          348          349 
## 8.629488e-05 1.203576e-03 5.342945e-05 8.276439e-05 4.172393e-03 
##          350          351          352          353          354 
## 2.010043e-04 1.813412e-04 1.103757e-03 5.116216e-04 1.421534e-03 
##          355          356          357          358          359 
## 3.237523e-07 2.124335e-03 1.656522e-05 2.107369e-03 1.064613e-06 
##          360          361          362          363          364 
## 1.406517e-04 4.036877e-05 1.898873e-04 7.213794e-04 1.674341e-02 
##          365          366          367          368          369 
## 3.894949e-04 5.131893e-03 4.280909e-04 2.882945e-04 1.268574e-03 
##          370          371          372          373          374 
## 6.128398e-04 4.653312e-03 3.966617e-03 4.178548e-04 7.446793e-04 
##          375          376          377          378          379 
## 1.399947e-02 7.238569e-04 6.688734e-04 8.117083e-06 1.548044e-03 
##          380          381          382          383          384 
## 1.528814e-03 1.666879e-04 3.347214e-04 2.698099e-03 1.396226e-03 
##          385          386          387          388          389 
## 9.286380e-05 2.205523e-05 2.483743e-05 9.129923e-05 3.195493e-06 
##          390          391          392          394          395 
## 1.689474e-03 1.323054e-05 5.653313e-03 1.081766e-03 1.099372e-05 
##          396          397          398          399          400 
## 6.544350e-03 2.091592e-03 1.572124e-03 1.877792e-03 3.741160e-05 
##          401          402          403          404          405 
## 9.321803e-05 9.272344e-05 4.160715e-03 4.927077e-05 2.306507e-04 
##          406          407          408          409          410 
## 9.357719e-05 2.022122e-05 1.968139e-03 7.278751e-04 2.329951e-05 
##          411          412          413          414          415 
## 3.854759e-04 5.634831e-06 3.591673e-04 4.942919e-03 1.053912e-03 
##          416          417          418          419          420 
## 9.012352e-04 9.246236e-06 3.098641e-03 3.988597e-03 3.768479e-03 
##          421          423          424          425          426 
## 3.350123e-05 1.248138e-02 1.320237e-03 3.436121e-05 4.610486e-05 
##          427          428          429          430          431 
## 7.284765e-05 2.678897e-04 1.200142e-04 3.945567e-04 5.042197e-04 
##          432          437 
## 5.055682e-04 9.536790e-04
#Create histogram
 
#A density plot of the residuals
plot(density(resid(model4))) 

#Create a QQ plot qqPlot(model, main="QQ Plot") #qq plot for studentized resid 
leveragePlots(model4) # leverage plots

#Collinearity
vif(model4)
##       sdata$toptim      sdata$tmarlow     sdata$tpstress 
##           1.296847           1.087557           9.791870 
##       sdata$gender sdata$intgenstress 
##          22.979404          27.390333
sqrt(vif(model4)) 
##       sdata$toptim      sdata$tmarlow     sdata$tpstress 
##           1.138792           1.042860           3.129196 
##       sdata$gender sdata$intgenstress 
##           4.793684           5.233577